I need to prove that: if $\left\| u \right\|_{H^{1}(\Omega)} \leq 1$, exists $\sigma$, K such that $$\left\| e^{\sigma |u(.)|^{2}} \right\|_{L^{2}(\Omega)} \leq K.$$
I've tried so far: If $u$ has this limitation, then $\left\| u \right\|_{L^{2}(\Omega)} \leq 1$, but with that, maybe I'll need: If $\int_{\Omega}|u|^{2}\leq 1,$ than exists $\epsilon$ such $|u|^{2} \leq \epsilon$, and I don't know if this is true.
